Hence, by differentiating \(\mathcal {L}\)  into \(N_i\)  and applying the Karush-Kuhn-Tucker (KKT) conditions [1]} we obtain
\(\begin{split}{\left\lbrace \begin{array}{ll}\frac{\partial \mathcal {L}}{\partial N_i} = \sum _{m \in \mathbb {M}_i} \frac{-2A_m}{N_i^3} + \left(\lambda _{1i} -\lambda _{2i}\right)+\mu =0~(\star ),\\\lambda _{1i}\ge 0, \lambda _{2i} \ge 0, \lambda _{1i} \left(N_i - kN\right)=0 , \lambda _{2i} N_i =0.\end{array}\right.}\end{split}\) 
